Problematics | Think of a number, repeat and recycle

This week's fare is relatively simple, although the puzzles are not exactly sitters.

Last week's puzzles were meant to be moderately difficult, but were probably tougher than I had intended going by the number of wrong answers. This week's fare is relatively simple, although the puzzles are not exactly sitters. The first one is based on a 'Think of a number' trick:

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#Puzzle 21.1.

You to Friend #1: Write down a three-digit number twice so that it becomes a six-digit number. For example, if you think of 123, write down 123123. Then pass on the slip to Friend #2.

Friend #2: Received a six-digit number from Friend #1.

You to Friend #2: Divide it by 13. Ignore the remainder if any. Write down the quotient on a new slip of paper and pass it on to Friend #3.

Friend #3: Number received from Friend #2.

You: Divide this by 7. Ignore the remainder if any. Write the quotient on a new slip of paper and hand it to Friend #4.

Friend #4: Received a number from Friend #3.

You: Divide that by 11. Like your friends before you, ignore the remainder if any.

Friend #4: Done, and written on a new slip. Whom do I pass this on to?

You: Back to Friend #1.

Friend #1: Received... What???

What is the mathematics that explains why Friend #1 sees a number that takes him by surprise?

#Puzzle 21.2.

Mailbox: Last week's solvers:

#Puzzle 20.1 (part 1)  Hi Kabir,  Speed of minute hand = 360°/hr; speed of hour hand = 30°/hr. Their relative speed = 360 – 30 = 330°/hr  To meet again, the relative distance has to be a full circle = 360°. The two hands meet again in 360/330 = 12/11 hr = 1 hr 5 minutes 27 seconds (seconds rounded to two digits). The exact times when they meet between midnight and noon are:  1:05:27 am, 2:10:55, 3:16:22, 4:21:49, 5:27:16, 6:32:44, 7:38:11, 8:43:38, 9:49:05, 10:54:33.  — Rahul Agarwal, Bay Area, California

#Puzzle 20.1 (part 2)   Hello!  Solution is 36 minutes past noon.  The hour and the reversing minute hands meet in 80 minutes, during which time the hour hand goes 360°/12 * 80/60 = 40°.  So, the minute hand goes (360 - 40)/80 = 4°/min in reverse. This is slower than a minute hand at normal speed (6°/min).  For the correct time to show, the reversing minute hand and the normal minute hand should be at the same place, Let that time be T. Then, 6T = 360 – 4T  Solving, we get T = 36 minutes after noon.  — Rohit Khanna, Noida

Hi Kabir, I took the help of Scrabble Word Finder online to choose appropriate words for this puzzle, but it was still a bit difficult... Please see the attached image. In fact, there is one more possibility where you replace BRAG with GRAB. — Prabhjot Singh, Delhi

As a matter of fact, Prabhjot, it turns out there are at least three alternative solutions. Dr Nakul Makkar of Noida goes with GRAV and BOW while Ravi and Rudra Sondhi, a father-and-son team from Gurgaon, go with BRAW and VOG. All these are dictionary words.  Among the wrong answers to the second part of #Puzzle 20.1, many assumed that the two hands must coincide when the watch gives the correct time. That's not necessary, as Rohit Khanna's solution above shows.

Solved both puzzles: Rohit Khanna (Noida), Sunita & Naresh Dhillon (Gurgaon), Anil Kumar Goyal (Delhi), Prakash Bhate (Mumbai), Rahul Agarwal (Bay Area, California), Ravi & Rudra Sondhi (Gurgaon), Geetansha Gera (Faridabad), Jasvinder Singh (Nabha), Yojit Manral (Mumbai), Dr Roona Poddar Gupta (Delhi)

Solved #Puzzle 20.2: Joita Bengeri, Gopal Menon (Mumbai), Sandeep Bhateja (Hoshiarpur), Shivika Gupta (Delhi), Nipun Bamania (Mumbai), Dr Nakul Makkar (Noida), Gayathri, Madhuri Patwardhan (Thane), Vinod Mahajan (Delhi), Amardeep Singh (Meerut), Shishir Gupta (Indore), Jaikumar Bhatia (Ulhasnagar, Thane), Musarrat Rai Handa (Faridabad), Biren Parmar (Bay Area, California).

Apologies to Gopal Menon (Mumbai) for missing his name in last week's acknowledgements.

Problematics will be back next week. Please send in your replies to problematics@hindustantimes.com.